Biological systems often need to remember their current state to produce reasonable responses for future decisions. To achieve this, a capability of maintaining transited states caused by transient stimuli is required. One of the mechanisms that can create such an effect is a feedback loop.

Feedback loops are processes that connects the output signal of the system back to their input signal. The concept and mechanism of feedback loops have been extensively investigated in various model and experimental systems, such as Turing’s model of pattern formation; as well as investigations of metabolic endproduct inhibition, metabolic oscillations, and transcriptional self-repression [1]. It has also been proved that the feedback loops may be useful as a framework for understanding intracellular signalling systems.

There are two main types of feedback loops: negative and positive (Figure 1). Negative feedback loops appear in almost all biological signalling pathways. They are typical characteristics of systems, where the inverted output signal is fed back to the input. Depending on its characteristics and initial conditions, a single negative feedback loop can create four distinct signalling functions: basal homeostat, output limiter, adaptation, and transient generator [1]. In contrary, a positive feedback loop is a major characteristic of a system, where the input is fed by the forward output signal. Positive feedback can amplify signalling responses, alter kinetics, or create bistable switches [1].

A single positive feedback loop between two molecules A and B can be subcategorized into mutual activation (A activates B and B activates A) and mutual inhibition (A inhibits B and B inhibits A). Despite the same nature of these two feedbacks, their steady-state characteristics are however different. In a positive feedback formed by a mutual activation, two molecules A and B show the same expression states (both A and B are on or off). On the other hand, in a mutual inhibitory positive feedback (MIPF), two molecules A and B show different expression states (one is on while the other is off). Such MIPFs can be found in biological processes where transitions between two different stable states are necessary.

Kim et al. [2], introduce a memory effect (ME) in a biological system as a capability of maintaining two opposing states which can be reversed by transient stimuli. Therefore, MIPFs can serve as ideal candidates to provide a ME in the system (Figure 2). Despite the ME be acquired by only one MIPF, in their paper Kim et al., explored a more biologically relevant situation, namely, the advantages of interlinked MIPF systems in realizing the ME by simulating them under a broad range of parameters (see Biomodels database entries BIOMD0000000179 and BIOMD0000000180). They considered three major types of MIPFs: single, asymmetrically-interlinked and symmetrically-interlinked. Examples of single and interlinked MIPF systems are presented in Figure 3.

For a protein in a cell to function as an activator or an inhibitor, it needs to be post translationally modified, for example, dimerization or phosphorylation, hence MIPF in the model was implemented as a sequential process of mRNA(R)→protein(P)→modified protein(P’) (Figure 4A). Inhibitory regulations are denoted as s1 and s2. The asymmetrically-interlinked MIPF system (Figure 4B) is constructed by adding an inhibitory link (s3) to the single MIPF system. Similarly, the symmetrically-interlinked MIPF system (Figure 4C) is constructed by adding an inhibitory link (s3) to the asymmetrically-interlinked MIPF system.

Furthermore, Kim et al. also checked the robustness of each cellular memory circuit with respect to parameter perturbations. The robustness can be measured by the area of its MR. They showed that the symmetrically-interlinked MIPF (corresponding to the largest MR - see Figure 5) is the most robust to parameter perturbations. In contrary, asymmetrically-interlinked MIPF in the same parameter range, loses its ME. Thus, the authors conclude that the ME of an MIPF system becomes more robust to parameter perturbations as the MR expands.

Finally, the authors show that the MR also defines the robustness of the ME with respect to external noises. They conclude that the ME becomes more robust to external noises as the MR expands.

Figure 5 (adapted from [2]): (A) The memory region of the asymmetrically-interlinked MIPF system on (s1, s2) parameter space. (B) The memory region of the symmetrically-interlinked MIPF system on (s1,s2) parameter space.